Logical equality

<h2 id="definition">Definition</h2>
<p>Logical equality is an <a href="/facts/Logical_operation/VkPhUs6M">operation</a> on two <a href="/facts/Logical_value/SY4e2w8l">logical values</a>, typically the values of two <a href="/facts/Proposition/0877MFMD">propositions</a>, that produces a value of <i>true</i> if and only if both operands are false or both operands are true.
</p><p>The <a href="/facts/Truth_table/4KsmArDW">truth table</a> of p EQ q (also written as p = q, p ↔ q, Epq, p ≡ q, or p == q) is as follows:
</p>

Logical equality<table><tbody><tr><th>p</th><th>q</th><th>p = q</th></tr><tr><td>0</td><td>0</td><td>1</td></tr><tr><td>0</td><td>1</td><td>0</td></tr><tr><td>1</td><td>0</td><td>0</td></tr><tr><td>1</td><td>1</td><td>1</td></tr></tbody></table>

<h2 id="alternative-descriptions">Alternative descriptions</h2>
<p>The form (<i>x</i> = <i>y</i>) is equivalent to the form (<i>x</i> ∧ <i>y</i>) ∨ (¬<i>x</i> ∧ ¬<i>y</i>).
</p><p>
  
    
      
        (
        x
        =
        y
        )
        =
        ¬
        (
        x
        ⊕
        y
        )
        =
        ¬
        x
        ⊕
        y
        =
        x
        ⊕
        ¬
        y
        =
        (
        x
        ∧
        y
        )
        ∨
        (
        ¬
        x
        ∧
        ¬
        y
        )
        =
        (
        ¬
        x
        ∨
        y
        )
        ∧
        (
        x
        ∨
        ¬
        y
        )
      
    
    {\displaystyle (x=y)=\lnot (x\oplus y)=\lnot x\oplus y=x\oplus \lnot y=(x\land y)\lor (\lnot x\land \lnot y)=(\lnot x\lor y)\land (x\lor \lnot y)}

</p><p>For the operands <i>x</i> and <i>y</i>, the <a href="/facts/Truth_table/4KsmArDW">truth table</a> of the logical equality operator is as follows:
</p>
<table><tbody><tr><th colspan="2" rowspan="2">                    x        ↔        y              {\displaystyle x\leftrightarrow y}  </th><th colspan="2">y</th></tr><tr><th>T</th><th>F</th></tr><tr><th rowspan="2">x</th><th>T</th><td>T</td><td>F</td></tr><tr><th>F</th><td>F</td><td>T</td></tr></tbody></table>
<h2 id="inequality">Inequality</h2>
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> that is known as a <i><a href="/facts/Field_(mathematics)/xAjAS4ko">field</a></i>.  For Boolean algebra, this means that the logical operation signified by "+" is not the same as the <a href="/facts/Inclusive_disjunction/jbI2EGlM">inclusive disjunction</a> signified by "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the <a href="/facts/Exclusive_disjunction/FHyr7hGQ">exclusive disjunction</a> signified by "XOR" or "⊕".  Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years.  At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality:
</p>

x
              
              
                
                +
                y
              
              
                x
              
              
                
                ≢
                y
              
              
                J
                x
                y
              
            
            
              
                x
              
              
                
                   
                  X
                  O
                  R
                   
                
                y
              
              
                x
              
              
                
                ≠
                y
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}x&+y&x&\not \equiv y&Jxy\\x&\mathrm {~XOR~} y&x&\neq y\end{aligned}}}

<p>This explains why "EQ" is often called "<a href="/facts/XNOR/LW793PwY">XNOR</a>" in the <a href="/facts/Combinational_logic/L1kaaVK5">combinational logic</a> of circuit engineers, since it is the <i>negation</i> of the <i><a href="/facts/XOR/FHyr7hGQ">XOR</a></i> operation; "NXOR" is a less commonly used alternative.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".
</p>
<h2 id="see-also">See also</h2>
<ul>
<li>Philosophy portal</li><li>Psychology portal</li></ul>
<ul><li><a href="/facts/Boolean_function/uT7E8pqC">Boolean function</a></li>
<li><a href="/facts/If_and_only_if/bYSxGJ66">If and only if</a></li>
<li><a href="/facts/Logical_equivalence/ysCWmsxA">Logical equivalence</a></li>
<li><a href="/facts/Logical_biconditional/AghYtFSh">Logical biconditional</a></li>
<li><a href="/facts/Propositional_calculus/MNt5v31V">Propositional calculus</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li> Media related to Logical equality at Wikimedia Commons</li>
<li>Mathworld, <a href="http://mathworld.wolfram.com/XNOR.html">XNOR</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Keeton, Brian; Cavaness, Chuck; Friesen, Geoff (2001), Using Java 2, Que Publishing, p. 112, ISBN 9780789724687. <a href="9780789724687" target="_blank">9780789724687</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Logical equality open-in-new

Logical equality